Signal Innovations Group proposes a hierarchical Bayesian approach for non-linear dimensionality reduction that addresses three key challenges: learning a reversible mapping from a high-dimensional observed space to a low-dimensional embedded space, learning the dimension of the embedded space, and generating new high-dimensional data for a given location in the embedded space. The proposed generative approach is statistical and jointly learns the probabilistic reversible mapping and the dimension of the embedded space. The proposed approach also enables new high-dimensional data to be embedded in a previously learned low-dimensional space. A hierarchical Bayesian method is also proposed to learn a non-linear dynamic model in the low-dimensional space, allowing joint analysis of multiple types of dynamic data, synthesis of new dynamic data in the low-dimensional space, and mapping synthesized data to the high-dimensional observation space. The models are designed to uncover the relevant characteristics and structure of data through non-linear dimensionality reduction, which enables a human analyst to identify and explore the characteristics in the low-dimensional manifold space and generate new unobserved high-dimensional data.